The metropolis Hasting algorithm decides whether to jump based on posterior probability. Namely, likelihood x prior.
Then it seems the density function that you choose (e.g., Poisson or Gaussian) can have quite some impact on the asymptotic posterior distribution we obtain from MCMC, because I can imagine there might be a case where for likelihood function L1, the parameter set $\theta_1$ has higher likelihood than parameter set $\theta_2$, whereas for the other likelihood function L2, the relation is reversed (i.e., $L1(\theta_1) > L1(\theta_2)$, but $L2(\theta_1) < L2(\theta_2)$.
Then $\theta_1$ would gain more density in the asymptotic posterior with L1, whereas $\theta_2$ would gain more density in the asymptotic posterior with L2. Different choice of likelihood function might give different asymptotic posterior. Am I right with this?